Abstract: Conventional approaches to the emergence of mixed from pure states are based on taking partial
traces. This approach has limitations for example in the treatment of
fermions and bosons where N-particle pure states contain irreducible
multi-particle correlations. A new approach
based on states on algebras of observables and their restrictions to
subalgebras is presented here. The subalgebra here refers to the subsystem
being observed. It agrees with the usual answers for bipartite systems of nonidentical particles,
but that is not the case in general. For example there
exist two fermion state vectors with Schmidt number 1 for which partial trace
gives 1 as entropy. The GNS approach instead gives zero, a very reasonable answer.
The GNS approach seems very general and can be applied for example to systems
obeying braid statistics.

Abstract: The topology and geometry of the gauge orbit space (space of physically
relevant configurations) are expected to play a
crucial role in the question of a nonzero mass gap for Yang-Mills theories.
Earlier attempts in developing this point of view
were vitiated by the discovery of the so-called spikes. I shall review the
spikes and the existence of a volume measure on
the gauge orbit space and relate it to the mass gap
for 3 dimensional gauge theories. More recent results
will also include the extension of these ideas to extended supersymmetric theories.

Abstract: We will review the use of topology to understand the physics of
quantum Hall systems and topological insulators. We then discuss
the concept of topological order in an exactly solvable 2-dimensional
spin-1/2 model, the Kitaev honeycomb model. Finally we present some
results from the ongoing work at IMSc on Hubbard type models which
have phases with topological order.

Abstract: Several combinatorial problems in physics, mathematics and computer
science lead to a natural generalisation of the partitions of integers
-- these are called higher-dimensional partitions and were first
introduced by MacMahon. Two-dimensional or plane partitions have a
nice generating function like the one due to Euler for usual
partitions. No simple generating function exists for dimensions
greater than two. We discuss two aspects of these partitions -- their
exact enumeration and their asymptotic behaviour. We show the
existence of transforms that show that the problem of exact
enumeration becomes somewhat easier than expected. The transforms come
with combinatorial interpretations and enable us to compute partitions
of all integers <=25 in any dimension. We also discuss the asymptotic
behaviour of these partitions focusing on the three-dimensional
(solid) partitions for concreteness.